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PH 301 Dr. Cecilia Vogel Lecture 2
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Review Outline matter waves probability, uncertainty wavefunction requirements Matter Waves duality eqns interpretation
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Probabilty and Normalization the probability of the particle being in a volume of space the probability of the particle being in all of space, should be 1 (100%) If the integral over all space =1, the wavefunction is “normalized” Only normalized wavefunctions can be used to find absolute probability NORMALIZATION
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Probabilty and Averages The average value of x can be found by averaging the possible values of x but some are more probable than others so the average is weighted by the probability density EXPECTATION VALUE
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Probabilty and Averages The expectation value of any function of x can be found similarly: EXPECTATION VALUE
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Uncertainty The uncertainty in x is a measure of the spread in possible values of x It is not measurement error nor lack of knowledge The wavefunction is really spread out over many x values like a water wave that strikes many points on the shore
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Uncertainty and Averages The uncertainty in x can be found as the root mean square (rms) deviation UNCERTAINTY DEF UNCERTAINTY CALCULATION The uncertainty can more easily be calculated using
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Uncertainty Example An electron in 1 st excited state of an infinite 1-D square well 1-nm long has a wavefunction that is zero outside the box and inside the box equal to (x in nm) The uncertainty can be calculated using Mathcad Mathcad
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Uncertainty Principle The uncertainty in position is not restricted Can be arbitrarily small But uncertainty in position and momentum can’t both be arbitrarily small
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Wavefunction Requirements Mathematically, a wavefunction can be any function, so long as it is normalized. BUT to describe a real physical particle the wavefunction must obey the laws of physics. The law of physics that applies to wavefunctions of non-relativistic particles is the Time Dependent Schroedinger Eqn
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TDSE The Time Dependent Schroedinger Equation: cannot be derived agrees with empirical observation describes the time evolution of a particle, given its environment (like F=ma for classical particles).
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TDSE The Time Dependent Schroedinger Equation in 1-D: The Time Dependent Schroedinger Equation in 3-D:
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Four Requirements The wavefunction of a physical particle 1.must obey TDSE 2.must be normalizable must be finite everywhere must approach zero as x, y, z approach ± ∞ 3.must be continuous no physical quantity should change by finite amount for an infinitesimal change in position 4.must have cont. first spatial derivative. anywhere V is finite actually a consequence of TDSE
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